Question

Evaluate the limit:
$\underset{x\to 0}{lim}\frac{\sin 3x+7x}{4x+\sin 2x}$

Answer 1

We have,

$\underset{x\to 0}{lim}\frac{\sin 3x+7x}{4x+\sin 2x}$

It becomes $\frac{0}{0}$ from.

Dividing numerator and denominator by $x,$ we get

$=\underset{x\to 0}{lim}\frac{\frac{\sin 3x}{x}+\frac{7x}{x}}{\frac{4x}{x}+\frac{\sin 2x}{x}}$

$=\underset{x\to 0}{lim}\frac{3\times \frac{\sin 3x}{3x}+7}{4+2\times \frac{\sin 2x}{2x}}$

$=\frac{3\times 1+7}{4+2\times 1}$
$[\because \underset{x\to 0}{lim}\frac{\sin x}{x}=1]$

$=\frac{10}{6}=\frac{5}{3}$

Therefore,

$\underset{x\to 0}{lim}\frac{\sin 3x+7x}{4x+\sin 2x}=\frac{5}{3}$